For numbers A, B, C, and D, subtract A from B, (or vice-versa; you
must be left with a whole number, not a negative one). Repeat with
B and C, C and D, and D and A. After about 6 steps, you will always
end up with 0000. The puzzle is to get as many steps as possible.

I asked my students to keep adding random integers from 1 to 100 until
the sum exceeded 100. We then found the average number of terms
added. The answer seems to be e. Why? The more we do it, the
closer we get.

What can you determine about the four variables in an equation, given information
about the factors of three of them? By decomposing positive integers with even
numbers of factors into products of primes, Doctor Greenie starts to unpack this
puzzle, case by case.

A student struggles to identify a pair of positive integers, given their sum as well as the
sum of their greatest common factor and least common multiple. Doctor Greenie
applies some algebra and factorization to turn the problem into a Diophantine
equation.

Can you prove that if you add the digits of any multiple of nine, then
add the digits of that result, and keep going, you eventually wind up
with 9? For example, 99 => 9 + 9 = 18 => 1 + 8 = 9. Why does it work?